Root Test for Series: Statement, Solved Examples

Cauchy’s Root test for series is used to test the convergence or divergence of an infinite series. It states that if lim_n→∞ a_n^1/n is less than 1 then the series ∑a_n converges and if the limit is greater than 1 then the series diverges. In this article, we will study Cauchy root test of a series with examples.

Root Test Rule

Statement: Let $\displaystyle \sum_{n=1}^\infty a_n$ be a series of positive real numbers. Suppose that

R = $\lim\limits_{n \to \infty} (a_n)^{1/n}$.

Then by root test we have the following.

• If R<1, then the series ∑a_n converges.
• If R>1, then ∑a_n diverges.
• If R=1, the root test is indecisive.

Root Test Solved Examples

$\boxed{\color{blue}\textbf{Q}1:}$ Test the convergence of $1+\dfrac{1}{2^2}+\dfrac{1}{3^3}+\cdots$.

Answer:

The given series = $\displaystyle \sum_{n=1}^\infty \dfrac{1}{n^n}$

Here, $a_n=\dfrac{1}{n^n}$.

Therefore,

R = lim_n→∞ $(a_n)^{1/n}$

= lim_n→∞ $\left( \dfrac{1}{n^n} \right)^n$

= lim_n→∞ $\dfrac{1}{n}$

= 0.

Since the limit R<1, the given series converges by root test.

$\boxed{\color{blue}\textbf{Q}2:}$ Test the convergence of the series $\displaystyle \sum_{n=1}^\infty a_n$ where a_n = $\dfrac{(1+n)^{n^2}}{n^{n^2}}$.

Answer:

R = lim_n→∞ $(a_n)^{1/n}$

= lim_n→∞ $\left( \dfrac{(1+n)^{n^2}}{n^{n^2}} \right)^n$

= lim_n→∞ $\left( \dfrac{n+1}{n} \right)^n$

= lim_n→∞ $\left( 1+\dfrac{1}{n} \right)^n$

= e >1 as the value of e lies between 2 and 3.

Since the limit R>1, the given series diverges by root test.

Related Articles:

• Convergence of a Series: Definition, Formula, Examples
• Comparison Test for Series: Statement, Examples [Limit Form]
• Ratio Test for Series: Statement, Solved Examples

Homework:

Test the convergence of the following series:

1. $\displaystyle \sum_{n=1}^\infty \dfrac{n^{n^2}}{(1+n)^{n^2}}$
2. $\dfrac{1}{2}+\dfrac{1}{3^2}+\dfrac{1}{4^3}+\cdots$

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